Why is exact inference in a Bayesian network both NP-hard and P-hard?
It's not completely clear from your question, but it looks like you want to prove that exact inference in a Bayesian Network is both NP-Hard and P-Hard.

It appears that you have proven that it is NP-Hard, but are unsure how to show that it is also P-Hard. 

This is more of a TCS question than an AI question, but shouldn't be too difficult. You just need to pick a P-Complete problem and reduce it to BN.